Challenge Level

This problem offers opportunities for pupils to explore number patterns in their own way. As well as applying number and calculation skills, learners will have the chance to make conjectures and reason mathematically as they try to explain the patterns they observe.

*This problem featured in the NRICH Primary webinar in April 2022.*

Ideally, you could introduce this task by taking your class on a short walk to observe house numbering in nearby streets. While many residential roads are numbered with odds on one side and evens on the other, this is not always the case so you may wish to check what your neighbouring streets are like beforehand. Ask learners what they notice about the numbering as you walk.

You can then facilitate a discussion about the children's observations and introduce the idea of representing the house numbers in a plan view. You can lead into the first part of the task by suggesting that as you walk down the street, you might add the house numbers in various ways. To give the group a flavour of this, begin with adding diagonal pairs across the street. Set children off in pairs exploring these sums, and then draw them back for a mini plenary. What have they noticed? Some will offer their observations about when the totals are the same, some may mention the totals increasing by the same amount as they move to the next group of four house numbers.

Let the class know that mathematicians like to be sure that a pattern they have observed will *always* be true. Can the children explain why their noticings will always be true? Give them time in pairs to talk about this and circulate so you can listen in on their discussions. You could warn a few pairs that you'd like them to share their thinking with the whole class in the
next mini plenary. Listen out for those who realise that consecutive odd numbers increase by 2, and so do consecutive even numbers. So, when adding diagonally across the street, although you have an odd that is bigger by 2 compared with its next-door odd, it is being added to an even that is less than 2 compared with its next-door even.

At this point, you can invite learners to consider an alternative way of adding the house numbers. You could show them some ideas from the problem, or you can encourage them to explore their own ideas. Leave enough time in the final plenary for a few pairs to report back on their investigation and the associated noticings, and explanations.

What do you notice about your answers?

Why does that pattern or patterns occur?

What could you explore now?

Some pupils may find digit cards useful to represent the house numbers. Calculators could also be helpful to free learners up to think creatively.

If it has not come up naturally, you could introduce the idea that some streets have terraces of three houses together:

and some have fours:

Learners can explore the addition of these groups too!